Abstract An n -dimensional lattice polytope $${\mathcal {Q}}_\sigma $$ Q σ can be associated to any composition $$\sigma $$ σ of a positive integer n , as a special case of constructions due to Pitman–Stanley and Chapoton. The entries of the h -vector of $$\sigma ,$$ σ , introduced by Chapoton, enumerate the lattice points in $${\mathcal {Q}}_\sigma $$ Q σ by the number of their nonzero coordinates. Chapoton conjectured that this vector is equal to the h -vector of a flag simplicial polytope. This paper proves this conjecture. Moreover, it shows that the gamma-vector associated to the h -vector of $$\sigma $$ σ is nonnegative by means of an explicit combinatorial interpretation and confirms certain other conjectures of Chapoton on the lattice point enumeration of composition polytopes. A combinatorial interpretation of their $$h^*$$ h ∗ -polynomials is deduced.

We classify trivalent graphs with 16 vertices and 16 edges that arise from intersecting two quadratic surfaces in tropical 3-space. There are 4,009 such graphs, representing maximally degenerate stable models of elliptic curves realized as tropical complete intersections of two quadrics. Our classification is derived from 405,246,030 regular unimodular triangulations of the 4-dimensional Cayley polytope.

Abstract Let $$X_1,\ldots , X_{d+2}$$ X 1 , … , X d + 2 be random points in $$\mathbb {R}^d$$ R d . The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $$P:= [X_1,\ldots , X_{d+2}]$$ P : = [ X 1 , … , X d + 2 ] , is a simplex. In the present paper, we study a refined version of this problem which asks to determine the probability that P has a given combinatorial type. It is known that there are $$\lfloor d/2\rfloor +1$$ ⌊ d / 2 ⌋ + 1 possible combinatorial types of simplicial d -dimensional polytopes with at most $$d+2$$ d + 2 vertices. These types are denoted by $$T_0^d, T_1^d, \ldots , T_{\lfloor d/2 \rfloor }^d$$ T 0 d , T 1 d , … , T ⌊ d / 2 ⌋ d , where $$T_0^d$$ T 0 d is a simplex with $$d+1$$ d + 1 vertices, while the remaining types have exactly $$d+2$$ d + 2 vertices. Our aim is thus to compute the probability $$ p_{d,m} := \mathbb {P}[P \text { is of type } T_{m}^d], \qquad m\in \{0,1,\ldots , \lfloor d/2 \rfloor \}. $$ p d , m : = P [ P is of type T m d ] , m ∈ { 0 , 1 , … , ⌊ d / 2 ⌋ } . The classical Sylvester problem corresponds to the case $$m=0$$ m = 0 . We shall compute $$p_{d,m}$$ p d , m for all m in the following cases: (a) $$X_1,\ldots , X_{d+2}$$ X 1 , … , X d + 2 are i.i.d. normal; (b) $$X_1,\ldots , X_{d+2}$$ X 1 , … , X d + 2 follow a d -dimensional beta or beta prime distribution, which includes the uniform distribution on the ball or on the sphere as special cases; (c) $$X_1,\ldots , X_{d+2}$$ X 1 , … , X d + 2 form a random walk with exchangeable increments. As a by-product of case (a) we recover a recent solution to Youden’s demon problem which asks to determine the probability that, in a one-dimensional i.i.d. normal sample $$\xi _1,\ldots , \xi _n$$ ξ 1 , … , ξ n , the empirical mean $$\frac{1}{n} (\xi _1 + \ldots + \xi _n)$$ 1 n ( ξ 1 + … + ξ n ) lies between the k -th and the $$(k+1)$$ ( k + 1 ) -st order statistics. We also consider the conic (or spherical) version of the refined Sylvester problem and solve it in several special cases.

Minimum energy configurations for interacting dipoles in simplex, orthoplex and hypercube crystals

It is shown that the smallest possible distance between two disjoint lattice polytopes contained in [ 0 , k ] 3 is exactly 1 2 ( 2 k 2 − 4 k + 5 ) ( 2 k 2 − 2 k + 1 ) for every integer k at least 4. The proof relies on modeling this as a minimization problem over a subset of the lattice points in the hypercube [ − k , k ] 9 . A characterization of this subset allows to reduce the problem to computing the roots of a finite number of degree at most 4 polynomials, using symbolic computation.

Abstract In 2021, Ordentlich, Regev, and Weiss made a breakthrough that the lattice covering density of any ‐dimensional convex body is upper bounded by , improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound , and this result was extended to certain symmetric convex bodies by Gritzmann. The constant above is independent on . In this paper, we show that such a bound can be achieved for more general classes of convex bodies without symmetry, including anti‐blocking bodies, locally anti‐blocking bodies and ‐dimensional polytopes with vertices.

Gathered under the name of metabolic networks, trophic, biochemical, and urban networks are here handled as a single field. In the Linear Inverse Modeling framework, these highly complex objects of research are all mathematically represented by weighted oriented graphs whose vertices are compartments and edges are flows (or flux) of matter or energy. Flows satisfying realistic metabolic constraints belong to very anisotropic high dimensional polytopes that cannot be analytically determined. Sampling the polytope of solutions yields a set of possible scenarios for the metabolic network. Different Markov Chain Monte Carlo (MCMC) algorithms together with their most recent implementations are scrutinized, leading to design an updated R package called {samplelim}. Comparison of the most recent implementations in terms of both computation time and sampling performances follows a methodology involving acknowledged and new statistical diagnostics and indexes. Application on real data metabolic networks of the three types shows that {samplelim} gathers the best properties of previous implementations of these MCMC algorithms. Code Repositories: The source code of the package {samplelim} is publicly available from its GitHub repository. 1 The code to reproduce computations has its own private GitHub repository. 2

Quint and Shubik conjectured that a nondegenerate [Formula: see text] game has at most [Formula: see text] Nash equilibria in mixed strategies. The conjecture is true for [Formula: see text] but false for [Formula: see text]. We answer it positively for the remaining case [Formula: see text], which had been open since 1999. The problem can be translated to a combinatorial question about the vertices of a pair of simple n-polytopes with 2n facets. We introduce a novel obstruction based on the index of an equilibrium, which states that equilibrium vertices belong to two equal-sized disjoint stable sets of the graph of the polytope. This bound is verified directly using the known classification of the 159,375 combinatorial types of dual neighborly polytopes in dimension five with 10 facets. Nonneighborly polytopes are analyzed with additional combinatorial techniques where the bound is used for their disjoint facets. Funding: This work was supported by the Deutsche Forschungsgemeinschaft Priority Program “Combinatorial Synergies” [Grant 539847176].

Abstract The gradient bounds of generalized barycentric coordinates (GBCs) play an essential role in the $H^{1}$ norm error estimate of generalized barycentric interpolations (Gillette, Rand & Bajaj (2012) Error estimates for generalized barycentric interpolation. Adv. Comput. Math., 37, 417–439.). Similarly, an $H^{k}$ norm error estimate, $k>1$, requires upper bounds of higher-order derivatives. Due to the nonpolynomial nature of GBCs, existing techniques for proving the gradient bounds do not easily extend to higher-order cases. In this paper, we propose a new method for deriving upper bounds of higher-order derivatives for the Wachspress GBCs on simple convex $d$-dimensional polytopes, $d\ge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of fourth- or higher-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.

Rectangulotopes

Finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets

Using the Kreuzer-Skarke database of 4-dimensional reflexive polytopes, we systematically constructed a new database of orientifold Calabi-Yau threefolds with h1,1(X) ≤ 12. Our approach involved non-trivial ℤ2 involutions, incorporating both divisor exchanges and multi-divisor reflections acting on the Calabi-Yau threefolds. Each proper involution results in an orientifold Calabi-Yau threefolds and we constructed 320, 386, 067 such examples. We developed a novel algorithm that significantly reduces the complexity of determining all the fixed loci under the involutions, and clarifies the types of O-planes. Our results show that under proper involutions, the majority of cases end up with O3/O7-plane systems, and most of these further admit a naive Type IIB string vacua. Additionally, a new type of free action was determined. We also computed the smoothness and the splitting of Hodge numbers in the ℤ2-orbifold limit for these orientifold Calabi-Yau threefolds.

The stochastic sandpile model (SSM) is a generalisation of the standard Abelian sandpile model (ASM), in which topplings of unstable vertices are made random. When unstable, a vertex sends one grain to each of its neighbours independently with probability $p \in (0,1)$. We study the SSM on complete graphs. Our main result is a description of the recurrent states of the model. We show that these are given by convex sums of recurrent states of the ASM. This allows us to recover a well-known result: that the number of integer lattice points in the $n$-dimensional permutation polytope is equal to the number of labelled spanning trees on $n$ vertices. We also provide a stochastic version of Dhar's burning algorithm to check if a given (stable) state is recurrent or not, which runs in linear time. Finally, we study a family of so-called partial SSMs in which some vertices topple randomly while others topple deterministically (as in the ASM, sending one grain to all neighbours). We show that this distinction is meaningful, yielding sets of recurrent states that are in general different from those of both the ASM and SSM. We also show that to get all recurrent states of the SSM, we can allow up to two vertices to topple deterministically.

From [Formula: see text]-dimensional reflexive polytopes one can construct [Formula: see text] complex-dimensional Calabi–Yau manifolds as hypersurfaces in toric varieties. In this contribution to the DANGER 3: Data Numbers and Geometry conference proceedings, we summarise previous work [P. Berglund, Y.-H. He, E. Heyes, E. Hirst, V. Jejjala and A. Lukas, New Calabi–Yau Manifolds from Genetic Algorithms (2023)] generating reflexive polytopes using genetic algorithms. As a proof of principle, we demonstrate that the genetic algorithm can reproduce known reflexive polytopes in two, three and four dimensions. Motivated by this result, we construct five-dimensional reflexive polytopes and establish that many of these are not in existing datasets and therefore give rise to new Calabi–Yau four-folds.

Small covers arising from 3-dimensional simple polytopes. The geometry of the tangent bundle of a three dimension small cover is an old topic. The theory is of great importance in mathematics and physics. It is an interesting question to understand whether tangential bundles exist on the 3-dimensional small cover. In this paper a three dimension small cover on the tangent bundle is defined by using a spin structure on the compact and oriented smooth three manifold. The SO(3) is diffeornorphic to the unit tangent bundle of the two-sphere. Since the SO(3) is diffeornorphic to RP3. The line bundle over RP3 is just trivial bundle. We use fiber bundle pullbacks measures prove that 3-dimensional small cover has a trivial tangent bundle. To the knowledge of the author of this paper, we only considered the special case on small cover and effectively connected Clifford algebras, spin groups, and small cover. Note that the general case is much more complicated. Some results in the general case were not proved.

We study the integer decomposition property of lattice polytopes associated with the $n$-dimensional smooth complete fans with at most $n+3$ rays. Using the classification of smooth complete fans by Kleinschmidt and Batyrev and a reduction to lower dimensional polytopes we prove the integer decomposition property for lattice polytopes in this setting.

In our paper [3], we classify simple regular polyhedral BP-complexes, which are polyhedral complexes satisfying certain natural conditions (B) and (P) on their vertex structures. Although 2-skeletons of higher dimensional regular polytopes are simple regular polyhedral complexes, they are not included in the above classification because they do not satisfy the condition (B), cf. [4]. It is natural to consider some suitable class of regular polyhedral complexes containing 2-skeletons of higher dimensional regular polytopes. In this paper, relaxing the condition (B), we investigate a non-degenerate regular polyhedral complex satisfying the condition (P), and give a partial answer to the classification of such polyhedral complexes.

Recursive constructions for the higher Stasheff—Tamari orders in dimension three using the Outer Tamari and Tamari Block posets

Abstract Let $P \subseteq {\mathbb {R}}^{d}$ be a polytope and let $\textbf {w}$ be an interior point of $P$. The coefficient of asymmetry$\operatorname {ca}(P,\textbf {w}):= \min \{ \lambda \geq 1: \textbf {w} - P \subseteq \lambda (P - \textbf {w}) \}$ of $P$ about $\textbf {w}$ has been studied extensively in the realm of Hensley’s conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We zero in on the coefficient of asymmetry for lattice zonotopes, that is, Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point for which the coefficient of asymmetry is bounded above by an explicit constant in $\Theta (d \log \log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills’ lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture (which, in fact, has been proved), and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry for certain lattice zonotopes.

We describe several analytical (i.e., precise) results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm.